Spread the insides of the rye bread with Russian dressing, about 1
tablespoon of dressing each. Distribute the corned beef evenly among
the sandwich bottoms. Top the corned beef with sauerkraut, then turkey,
then Swiss cheese. Place the remaining bread slices on top.

If you want cold sandwiches, stop there. But if you want Reuben Sandwiches on the heavenly side, grill them, panini-style:

On a large griddle (or in batches in a skillet):
Butter one (outer) side of each sandwich, using about 1 teaspoon of
butter. Place them buttered-side-down on the griddle or skillet and turn
the heat to medium. Cook on one side until golden brown, pressing them
with a spatula now and then. Butter the other, dry side of the
sandwiches and turn them over. Weight the sandwiches with a heavy plate
or frying pan, whatever, and cook the second side until golden.

In a panini maker: Butter both sides of the sandwiches and pop them into the panini maker. Cook just until golden brown and a little crispy.